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I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.
Yes, some clean-up would be nice. I replied to the queries.
In that case, Zoran’s point stands.
Looks like the remark in question was originally added by Ronnie Brown over a year ago.
There is still left-over query box discussion at compactly generated space between Zoran, Todd, Mike and Toby . It looks to me like all issues have been clarified there. Does anyone feel like brushing up the entry?
Actually, looking at it again I see maybe an interpretation of the remark that might make it make sense. If by $k Top(X,Y)$ we mean a specific topology which is not necessarily a k-space, then cartesian closure of the category of k-spaces and continuous maps would be about a homeomorphism relating to the k-ifications of this $k Top(X,Y)$ when X and Y are k-spaces, while cartesian closure of the category of all spaces and k-continuous maps could be phrased either as saying the same thing (invoking the fact that they are equivalent) or about a k-homeomorphism relating to $k Top(X,Y)$ itself rather than its k-ification. And in the latter case saying that it is actually a homeomorphism rather than just a k-homeomorphism would be saying something stronger, relating to a characterization of $k Top(X,Y)$ as an arbitrary space rather than (its k-ification) as a k-space. Is this making sense?
Unfortunately, I am too much in a hectic mode before the long travel next week, so I can not delve into this discussion until about next Thursday…
I’ve edited the entry to clarify some, by distinguishing notationally between the category of k-spaces and continous maps, and the category of all spaces and k-continuous maps. Although they are equivalent, the question at issue seems to revolve around their non-identical-ness.
I still do not have enough time to delve into this but I was told by my student that
is extremely clear about when the weakly Hausdorff assumption is needed and when not. I quoted it in $n$Lab.
Thanks for the reference; I had a look at it and tried to clarify the entry a trifle further.
compactly generated space says that a topological space is compactly generated iff
$X$ is an identification space of a disjoint union of compact Hausdorff spaces.
It is clear to see that a quotient space of a disjoint union of compact Hausdorff spaces if compactly generated, namely every compact Hausdorff space is compactly generated, disjoint union of compactly generated Hausdorff is compactly generated Hausdorff and any quotient of a compactly generated Hausdorff space is compactly generated. Conversely, there are many ideas of covers of $X$ by compact Hausdorff spaces, one needs to choose good enough such that the definition for compactly generated (not necessarily even weakly Hausdorff) space can be tested only on the morphisms from the cover: then one does the identification by identifying points with the same image. However, any attempt which I tried in a quick attempt use the axiom of choice, most often within a proper class or something of the sort. What is the clean set-theoretic way to do it ?
What makes you think there is one? (-:
So what the entry claims ? It says that it is a characterization of a compactly generated space. Even if we use the axiom of choice is it within a set, or we entail a change of universe or what ? Something is claimed and my attempts to understand it get into set-theoretical nightmare, if I want a full generality.
Well, I don’t know the answer. I was just saying, just because something is true doesn’t necessarily mean it can be proven in a set-theoretically clean way. (-:
I felt that :)
I have reorganized the sections at compactly generated space a little (check if you agree that it is better now), added a stubby Examples-section and a reference.
By the way, there is still lots of query-box discussion there. Maybe it can be removed or else turned into definite content?
I moved the (still empty) section on “weak Hausdorffification” to weakly Hausdorff space. Maybe k-space and compactly generated space should be separate pages too?
In compactly generated topological space there is an inconclusive discussion box about local cartesian closure of the category of compactly generated spaces. I want to bring your attention to this paper I have just stumbled upon
Booth, Peter I. The exponential law of maps. II. Math. Z. 121 (1971), 311–319
where the author claims that the category of compactly generated spaces over a Hausdorff space is cartesian closed. Unfortunately, for the definition of the internal hom-object he refers to another paper of his where the definition is stated in a different language (and involves further references) and I find it difficult to put the definition and the proof together. Perhaps someone will be motivated enough to take a closer look and decide whether the proof is valid.
I have added in, here, the argument for why every CW-complex is compactly generated.
Returning to #18: I’m fairly skeptical of local cartesian closure. This paper for example says that the category of Kelley spaces (meaning compactly generated Hausdorff spaces) is not locally cartesian closed.
In the entry where it says “by Yoneda lemma arguments” I have added a pointer to the actual argument (prop.).
Regarding #5 and #6: I vote for simply giving some name such as $[X,Y]$ to the actual exponential object in compactly generated spaces and then simply saying that there is a homeomorphism $[X,[Y,Z]]\simeq [X \times Y, Z]$.
I don’t see how the present state of the entry is trying to improve on this simple fact that the entry should state.
Comment #21 should point here.
Thanks. Sorry.
I don’t have the time to re-understand what was going on in #5-6, but I agree that the simple and important facts should be stated up-front in clean notation.
There are some old queries on that page that could probably be discarded…
I just noticed that the section Local Cartesian closure contained the following query box exchange (now moved from there to here) which remains unresolved:
+– {: .query}
Mike Shulman: What precisely does “get an exponential law” mean? Do you mean that $k Top/B$ is cartesian closed if $B$ is $T_0$?
Toby: Hopefully that is explained in the new article.
Mike: Which new article? exponential law for spaces? That page doesn’t talk about fibered exponentials at all.
Toby: Seeing this later, I no longer know what article I meant.
=–
I expect that P. Booth would be an author of such an article. It might be this by Booth and Brown. I can probably firm this up soon.
@Todd here’s a better link: https://doi.org/10.1016/0016-660X(78)90049-1, title is Spaces of partial maps, fibred mapping spaces and the compact-open topology, if that rings any bells for anyone.
Thank you David.
If in doubt ask Ronnie Brown or Peter Booth. At that time (later 1970s early 80s) a lot was done on partial maps.
Removed an old query box:
Zoran Škoda: I do not understand the remark. I mean if the domain is k-space then by the characterization above continuous is the same as k-continuous. Thus if both domain and codomain are continuous then homeo is the same as k-homeo. I assume that even in noHausdorff case, the test-open topology for $X$ and $Y$ k-spaces gives a k-space and that the cartesian product has the correction for the k-spaces.
Todd Trimble: That may be just the point: that the domain is not necessarily a $k$-space. I have to admit that I haven’t worked through the details of this exposition, but one thing I tripped over is the fact that we’re dealing with all topological spaces $X$, $Y$, not just $k$-spaces.
Mike Shulman: But any topological space is isomorphic in $k\Top$ to its $k$-ification, right? So $k\Top$ might as well be defined to consist of $k$-spaces and continuous maps.
Todd Trimble: Okay, you’re right that makes sense. So in that case, it seems that Zoran definitely has a point here.
Mike Shulman: See the nForum discussion.
[Administrative note: #1 - #24 were originally from another thread with name ’compactly generated space’. Thank you to Dmitri Pavlov for pointing out that the threads should be merged; I have deleted Dmitri’s comment requesting this to avoid confusion in the future, as it now would have been placed out of sync.]
added pointer to
both for the claim that the category is not locally Cartesian closed (which was already mentioned, but without reference (nor proof)) and for the claim that it is regular (which I added, thereby cross-linking the comment we had all along at regular category)
Is there any subcategory of topological spaces that is known to be locally cartesian closed? For example, are Δ-generated topological spaces locally cartesian closed?
Not as far as I know (other than the subcategory of discrete spaces). In this respect supercategories (of subcategories) of topological spaces are much better behaved. A nice example is Johnstone’s topological topos, which embeds all sequential topological spaces. Larger things like that are the new condensed sets and pyknotic sets, which embed compactly generated spaces (which generalize sequential spaces).
The are also the much older quasi-topological spaces which form a quasi-topos modulo your size-view preferences. (And the difference between condensed sets and pyknotic sets is a size preference (in each case).)
The entry claims that the category of compactly generated spaces is discussed in
On which page?
I haven’t found it yet, neither scanning through the table of contents nor searching an electronic copy for various choices of related keywords.
added pointer to:
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(with focus on compactly generated topological G-spaces)
added pointer to:
added pointer to:
as the apparent source of the notion of the “weak Hausdorff”-condition (attributed there to John C. Moore).
I have added pointer for the use of “k-ification”.
Our entry also claims the use of “kaonization” and attributes it to Postnikov, but I see almost no usage of “kaonization” and find nothing by Postnikov on the subject. What’s the intended reference?
It’s used in Quantum extensions of ordinary maps, but no information there as to source.
Yes, that’s the one hit I got which made me write “almost”. But this being from 2018 and not giving a citation, I am inclined to suspect that this author took the terminology from the nLab…
Also, the term “kaonization” doesn’t really make sense, does it. Even if Postnikov should have used this term somewhere (for which it would still good to have a reference) it might be a good idea not to advertize this usage further.
Presumably it was a joke form of ’k-ification’, like ionization, but its passing won’t be missed. Not that k-ificiation is elegant either.
Yes. “k-ification” is not elegant, but at least it makes sense as a translated shorthand for “Kompaktifizierung” (which is a standard term, albeit imprecise here), or, rather, something like “Kompaktgenerierung” (which is more precise albeit not standard, but might as well be if history had taken different turns).
I have definitely seen “kaonization” before. Can try to look up some references later. Concerning etymology, it could be a pun of “canonization”.
Okay, thanks. No rush, but if you have references, I’d be interested.
Found it in Postnikov’s book, and I think Postnikov might be the originator of this terminology. http://libgen.rs/book/index.php?md5=34A8C3C956EB80877F4E3CF5A297F514
On page 34 he talks about kaonic spaces (каонные пространства), defined as topological spaces X for which C⊂X is closed if and only if f^* C⊂K is closed for any continuous K→X if K is compact.
On page 33 he talks about kaonic maps (каонные отображения), defined as maps of sets f:X→Y for which K→X→Y is continuous whenever K→X is continuous and K is compact.
On page 387 he uses this terminology again.
Postnikov also uses the same terminology more extensively (over a dozen occurrences) in the book http://libgen.rs/book/index.php?md5=4BF450585846A0531FF485E34D062C0A.
Another source I was able to find is Postnikov’s translation of the book by Gabriel and Zisman into Russian: http://libgen.rs/book/index.php?md5=84F649B0DEE2C53F4101735ABE4ED8BE
Thanks!
I have added these pointers to the entry (here).
Do you see if, apart from “kaonic spaces” and “kaonic maps” he speaks of “kaonization”?
Do you know if any of this made it into English publications?
By the way, while we are hunting references: We are also still looking for a reference that would reflect Kelley’s “extensive” study of CG-spaces, as claimed by the entry. The reference Kelley 1955 seems to have only a single page on them, with just the basic definitions.
Re #55: In a sense it did, given that Andre Kornell does use “kaonization” extensively in his published papers. I could not find specific occurrences of “каонизация” in Postnikov’s writings, though.
Concerning “Kelley spaces”: this terminology appears to originate in Calculus of Fractions and Homotopy Theory by Gabriel and Zisman, where it is used all over the place.
They do not give any references, but I presume they refer to his book General Topology, which is one of the earliest sources for them.
Kelley does perform nontrivial things with k-spaces, see, for example, his version of the Arzela-Ascoli theorem for k-spaces on page 234 of his book.
Thanks again!
This made me break up the discussion of terminology into two items
terminology for the spaces themselves – now here.
terminology for the reflection operation – now here.
Postnikov is now cited for the former, no longer for the latter.
Kornell does use “kaonization” extensively in his published papers.
For better or worse. But how extensively really? Google gives me a single hit: arXiv:1811.01922, and that is – let’s beware – from over 9 years after the nLab started to – apparently incorrectly – proclaim this terminology in rev 8 (from Zoran).
It seems likely to me that an author who starts looking into the subject only from after 2009 will have picked up that terminology from the $n$Lab, where it was (until I finally changed it the other day) no less but a section title.
I fear we may just perpetuate a mistake that we were responsible for in the first place if we give any weight to “kaonization”.
But please let me know how extensively you feel Kornell has really been using it, meanwhile.
Re #57: “Kaonization” in Kornell’s paper may well be an example of a citogenesis, just like “étalé space” in Carchedi’s paper. We could ask him.
Concerning Zoran’s contribution to Revision 8, can we ask him directly?
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Hi Martin,
I have taken this verbatim from the comment you made public in #58. Do you want to keep that comment #58 public? If not, you can edit/clear it any time by clicking “edit” in the top right.
Do we have another source to use as the citation for the claim that Hurewicz gave the definition in lectures in Princeton 1948?
Does anyone know how to contact David Gale? (Searching for him seems almost impossible due to prominent namesakes.) If I had an email address, I would contact him to get this info first hand.
Woops. Thanks for catching. As you noticed, I had this mixed up already in the lecture notes statement pointed. Thanks for fixing!
Okay, thanks. It looks like, with your help, we saved this information last-minute from disappearing into forgottenness.
By the way, I used to assume that this David Gale is not the David Gale who wrote about k-spaces – but since you said above that our David Gale passed away in 2008, maybe that’s the same person after all? Do you know?
Okay, I found the proof: This “MacTutor biography” clarifies it. It’s the only source that I have seen so far which admits that this David Gale authored “Compact sets of functions and function rings”.
Okay, great, have created a category:people
-entry David Gale now.
I have tried to give more logical structure to the (old) section on the coreflection (now here) by adding numbered environments and cross-pointers, and disentangling the construction of the coreflection (which is still lacking a word on why it’s actually a coreflection) from the discussion of the equivalence $k Top \simeq Top_k$.
In the course of doing this I ended up doing similar edits to the Defintion section here (adding numbered environments and cross-pointers).
Finally, I have started a new Properties-section “Reflection into weak Hausdorff spaces” (here), showing the other adjunction.
In doing so, I have stuck – for the time being – with the notation “$k Top$” long used in the previous section, and so now $Top_{CGWH}$ is denoted “$h k Top$” here. I am not saying this is a good idea, necessarily, but it’s momentarily the least disruptive to the old material in the entry.
I have started a new Properties-subsection “Relation to locally compact Hausdorff spaces” (here)
Regarding the statement that a Hausdorff space is a k-space iff it is the quotient of a locally compact Hausdorff space:
Is that quotient then also the coequalizer in weak Hausdorff spaces (?) Need to think about this tomorrow when I am more awake…
$\,$
also, I have split the References-section into two subsections, and re-organized slightly, for more systematics
added this pointer, for relation to $\Delta$-spaces:
Added (here) pointer to
and grouped this together with Escadro-Lawson & Simpson 2004 and with Gaucher 2009 under “discussion in the generality of subcategory-generated spaces, including $\Delta$-generated spaces”.
I have expanded the statement of regularity (here) by saying that not only compactly generated Hausdorff spaces but also cg weakly Hausdorff spaces form a regular category.
(This is from the same page 3 in Cagliari, Matovani and Vitale 1995, in the Remark below the Theorem.)
I have added the references on (aspects of) local cartesian closure
Peter I. Booth, Ronnie Brown, Spaces of partial maps, fibred mapping spaces and the compact-open topology, General Topology and its Applications 8 2 (1978) 181-195 $[$doi:10.1016/0016-660X(78)90049-1$]$
Peter I. Booth, Ronnie Brown, On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps, General Topology and its Applications 8 2 (1978) 165-179 $[$doi:10.1016/0016-660X(78)90048-X$]$
Peter May, Johann Sigurdsson, §1.3.7-§1.3.9 in: Parametrized Homotopy Theory, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, arXiv:math/0411656, pdf)
Then I replaced (in the paragraph here) the line which was vaguely referring to May&Sigurdsson with concrete pointers to page and verse in these references.
I think it is still true that Fréchet spaces (i.e. the tvs ones) are compactly generated, since they are metrisable, hence first countable, hence sequential. Just an unfortunate clash of naming there.
Re #84: thanks. Some of that should be incorporated more explicitly. Let me get to that in a bit.
Re #84/#85: any indiscrete space with more than one point is also a (non-T₁) counterexample; $\alpha \mathbf{Q}$ shows that even weak Hausdorffness (in particular KC-ness or T₁-ness) is not enough.
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